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Quadratic and cubic Gaudin Hamiltonians and super Knizhnik-Zamolodchikov equations for general linear Lie superalgebras

Bintao Cao, Wan Keng Cheong, Ngau Lam

TL;DR

The paper advances the Gaudin model by extending diagonalization results to quadratic and cubic Gaudin Hamiltonians for general linear Lie superalgebras ${ rak{gl}}(p+m|q+n)$ and ${ rak{gl}}(m|n)$. It leverages super duality and truncation to transfer spectral information from large finite-rank Lie algebras ${ rak{gl}}(r+k)$ and ${ rak{gl}}(k)$ to the super setting, establishing simultaneous diagonalization on singular weight spaces and constructing corresponding eigenbases. It also formulates and analyzes (super) Knizhnik–Zamolodchikov equations in these contexts, proving stability of singular solutions under truncation and providing linear isomorphisms between solution sets across the categories. The results unify quadratic and cubic Gaudin theories across super and non-super cases, with explicit connections between KZ equations and their super-analogues, and offer practical transfer principles for eigenstructure from classical to superalgebra Gaudin models. This framework broadens the applicability of Gaudin-based spectral methods in representation theory and mathematical physics.

Abstract

We show that under a generic condition, the quadratic Gaudin Hamiltonians associated to $\mathfrak{gl}(p+m|q+n)$ are diagonalizable on any singular weight space in any tensor product of unitarizable highest weight $\mathfrak{gl}(p+m|q+n)$-modules. Moreover, every joint eigenbasis of the Hamiltonians can be obtained from some joint eigenbasis of the quadratic Gaudin Hamiltonians for the general linear Lie algebra $\mathfrak{gl}(r+k)$ on the corresponding singular weight space in the tensor product of some finite-dimensional irreducible $\mathfrak{gl}(r+ k)$-modules for $r$ and $k$ sufficiently large. After specializing to $p=q=0$, we show that similar results hold as well for the cubic Gaudin Hamiltonians associated to $\mathfrak{gl}(m|n)$. We also relate the set of singular solutions of the (super) Knizhnik-Zamolodchikov equations for $\mathfrak{gl}(p+m|q+n)$ to the set of singular solutions of the Knizhnik-Zamolodchikov equations for $\mathfrak{gl}(r+k)$ for $r$ and $k$ sufficiently large.

Quadratic and cubic Gaudin Hamiltonians and super Knizhnik-Zamolodchikov equations for general linear Lie superalgebras

TL;DR

The paper advances the Gaudin model by extending diagonalization results to quadratic and cubic Gaudin Hamiltonians for general linear Lie superalgebras and . It leverages super duality and truncation to transfer spectral information from large finite-rank Lie algebras and to the super setting, establishing simultaneous diagonalization on singular weight spaces and constructing corresponding eigenbases. It also formulates and analyzes (super) Knizhnik–Zamolodchikov equations in these contexts, proving stability of singular solutions under truncation and providing linear isomorphisms between solution sets across the categories. The results unify quadratic and cubic Gaudin theories across super and non-super cases, with explicit connections between KZ equations and their super-analogues, and offer practical transfer principles for eigenstructure from classical to superalgebra Gaudin models. This framework broadens the applicability of Gaudin-based spectral methods in representation theory and mathematical physics.

Abstract

We show that under a generic condition, the quadratic Gaudin Hamiltonians associated to are diagonalizable on any singular weight space in any tensor product of unitarizable highest weight -modules. Moreover, every joint eigenbasis of the Hamiltonians can be obtained from some joint eigenbasis of the quadratic Gaudin Hamiltonians for the general linear Lie algebra on the corresponding singular weight space in the tensor product of some finite-dimensional irreducible -modules for and sufficiently large. After specializing to , we show that similar results hold as well for the cubic Gaudin Hamiltonians associated to . We also relate the set of singular solutions of the (super) Knizhnik-Zamolodchikov equations for to the set of singular solutions of the Knizhnik-Zamolodchikov equations for for and sufficiently large.
Paper Structure (18 sections, 41 theorems, 88 equations)

This paper contains 18 sections, 41 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General linear (super)algebras
  4. Dynkin diagrams
  5. Central extensions
  6. Super duality
  7. Unitarizable modules over $\overline{\mathcal{G}}[q,m]_{(p,n)}$
  8. Quadratic Gaudin Hamiltonians and super KZ equations for $\widetilde{\mathfrak g}$, ${\mathfrak g}_{(p,n)}$ and $\overline{\mathfrak g}[q,m]_{(p,n)}$
  9. Quadratic Gaudin Hamiltonians on modules over $\widetilde{\mathfrak g}$, ${\mathfrak g}_{(p,n)}$ and $\overline{\mathfrak g}[q,m]_{(p,n)}$
  10. Super Knizhnik-Zamolodchikov equations for $\widetilde{\mathfrak g}$, ${\mathfrak g}_{(p,n)}$ and $\overline{\mathfrak g}[q,m]_{(p,n)}$
  11. Quadratic Gaudin Hamiltonians and super KZ equations for $\mathcal{G}_{(p,n)}$ and $\overline{\mathcal{G}}[q,m]_{(p,n)}$
  12. Quadratic Gaudin Hamiltonians for $\mathcal{G}_{(p,n)}$ and $\overline{\mathcal{G}}[q,m]_{(p,n)}$
  13. Diagonalization and correspondences between eigenbases
  14. Super Knizhnik-Zamolodchikov equations for $\mathcal{G}_{(p,n)}$ and $\overline{\mathcal{G}}[q,m]_{(p,n)}$
  15. Cubic Gaudin Hamiltonians for ${\mathfrak{gl}}(m|n)$
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\overline L^\otimes=\overline{L}_1 \otimes \cdots \otimes \overline{L}_\ell$ be a tensor product of unitarizable highest weight $\overline{\mathcal{G}}[q,m]_{(p,n)}$-modules. (For $m=p=0$, $\overline L_i$ are infinite-dimensional unitarizable highest weight modules over $\mathfrak{gl}(q+n)$). F

Theorems & Definitions (65)

  • Theorem 1.1: Theorem \ref{['thm:quad-diag']}
  • Theorem 1.2: Theorem \ref{['thm:cubic-diag']}
  • Theorem 1.3: Theorem \ref{['iso-KZ']}
  • Remark 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Proposition 2.7
  • ...and 55 more